A molecular reconstruction theorem for $H_{\omega }^{p(\cdot )}({\mathbb{R}}^n)$

In this article we give a molecular reconstruction theorem for $H_{\omega }^{p(\cdot )}({\mathbb{R}}^n)$. As an application of this result and the atomic decomposition developed in [5] we show that classical singular integrals can be extended to bounded operators on $H_{\omega }^{p(\cdot )}({\mathbb{R}}^n)$. We also prove, for certain exponents $q(\cdot )$ and certain weights $\omega$, that Riesz potential $I_{\alpha }$, with $0<\alpha <n$, can be extended to a bounded operator from $H_{\omega }^{p(\cdot )}({\mathbb{R}}^n)$ into $H_{\omega }^{q(\cdot )}({\mathbb{R}}^n)$, for $\frac{1}{p(\cdot )}:=\frac{1}{q(\cdot )}+\frac{\alpha }{n}$.